Problem 82 - Elementary Algebra Within Reach

Question

In Exercises 79-84, evaluate the expression. $$ \frac{7}{8} \div \frac{3}{16} $$

Step-by-Step Solution

Answer

\(\frac{14}{3}\)

Step 1: Convert Division to Multiplication

The division of fractions can be converted to a multiplication problem by multiplying by the reciprocal of the divisor. Reciprocal of a fraction is achieved by simply swapping the numerator and the denominator. Thus, \( \frac{7}{8} \div \frac{3}{16}\) is equivalent to \(\frac{7}{8} \times \frac{16}{3}\).

Step 2: Perform Multiplication

Now, proceed to multiply the numerators together and the denominators together. This yields \(\frac{7 \times 16}{8 \times 3}\), which simplifies to \(\frac{112}{24}\).

Step 3: Simplify the Fraction

Finally, simplify the fraction \(\frac{112}{24}\) by dividing the numerator and denominator by their greatest common divisor, which is 8. This results in \(\frac{14}{3}\).

Key Concepts

Multiplying Fractions

When it comes to multiplying fractions, the process is straightforward and can be summed up in a few easy steps. To multiply two fractions, follow these simple rules:

• Multiply the numerators together. The numerator is the top part of the fraction. For example, in our problem, you multiply 7 and 16.
• Multiply the denominators together. This is the bottom part of the fractions. In the exercise, it’s 8 and 3.

Therefore, multiplying will give you a new fraction: \[\frac{7 \times 16}{8 \times 3} = \frac{112}{24}.\]And that's how you multiply fractions! It's like scaling the parts of your fractions together, leading you to a whole new fraction.

Reciprocal of a Fraction

The concept of the reciprocal of a fraction is crucial in division problems involving fractions. A reciprocal is simply a flipped version of an original fraction. This means that the numerator and the denominator exchange places.

For example, the reciprocal of the fraction \(\frac{3}{16}\) is \(\frac{16}{3}\).By finding the reciprocal, you transform the division problem into a multiplication one.What once was \(\frac{7}{8} \div \frac{3}{16}\)is rewritten as \(\frac{7}{8} \times \frac{16}{3}.\)

• Remember, every whole number has a reciprocal too. For example, the reciprocal of 5 is \(\frac{1}{5}.\)
• Fraction reciprocals make dividing fractions as easy as multiplying them.

Simplifying Fractions

Simplifying fractions is a valuable skill to ensure that your answer is as clean and simple as possible. To simplify a fraction, you must reduce it to its smallest possible version without changing its value. The key is to find a common number that divides both the numerator and the denominator.
For example, in \(\frac{112}{24},\)both numbers are divisible by 8, simplifying the fraction to \(\frac{14}{3}.\)Here are some steps to simplify fractions:

• Identify the greatest number that divides both the top and bottom numbers evenly.
• Divide both the numerator and the denominator by this number.
• The result is a simplified fraction, where no further simplification is possible.

Greatest Common Divisor

The greatest common divisor (GCD) is the highest number that divides two or more numbers without leaving a remainder. It plays a significant role in simplifying fractions because it determines how much you can reduce your fraction by.
To find the GCD of two numbers, consider \(112\)and \(24.\)The GCD is 8 since that is the largest number they both can be evenly divided by.A simple way to find the GCD is:

• List the factors of each number.
• Identify the largest common factor.

Using the GCD, you can quickly reduce complex fractions to their simplest form, turning \(\frac{112}{24}\)into \(\frac{14}{3}.\) This makes the resulting fraction easier to understand and work with.